These days "test-time scaling" mostly means letting the model talk to itself for longer, but the first genuinely surprising results came from plain sampling. Google's AlphaCode generated millions of candidate programs and filtered them down to a handful of submissions, which beat the average human programmer in 2022, before ChatGPT even showed up.
Sampling is what AI is good at. Making examples and doing LeetCode are similar in that verification is clear and cheap. Compared to that, "proof" is still a vague concept, except where Lean works. See the fuss over the ABC conjecture. So humans are still needed.
The interesting question to me is what happens after enough learning from "sampling." Isn't AlphaGo's move 37 an AI's nose? If that happens in mathematics, we may end up with results that are correct, machine checkable, and not explainable in any way we find satisfying.
Interesting read!
https://ncatlab.org/nlab/files/why_abc_is_still_a_conjecture...
Wikipedia is maybe the narrow end of a wedge into this topic but the controversy revolves around a very large and very complex paper that few people are equipped to understand and some of those who are able believe the proof is false.
It doesn't help that the author is rather withdrawn and not willing to spend any effort in making it more approachable.
Some tried, and said they found gaps in the proof, to which the author responded, but they were not convinced.
And that's essentially the situation since 2018.
The trick is avoiding the infinite monkey problem. If your problem is amenable to RL, then you probably don't even need an LLM, Monte Carlo Tree Search gets you there with less expensive hardware.
Agreed. I find that after seeing these results from OpenAI we undeniably have a machine that has:
* General knowledge of nearly every subject humanity has ever learned
* The ability to simulate reasoning (albeit sometimes not very well) with that knowledge
* The ability to reference across the domains of knowledge
To me, this is more or less what I would think "Artificial General Intelligence" is. It's the cumulative knowledge of all general human intelligence, baked into an artificial form, which can then use that knowledge to achieve novel goals.
In many cases of mathematical breakthroughs there is an insight that comes from just happening to know a combination of already existing ideas and then combining them to solve that problem. This is where having that general knowledge seems particularly strong because we can run these machines for weeks on end effectively trying to brute force.
That being said, I could never imagine an LLM in its current form inventing something as elegant as the Fourier transform.
So then you need to explain ARC-AGI-3: https://arxiv.org/abs/2603.24621
"Our testing shows humans can solve 100% of the environments, in contrast to frontier AI systems which, as of March 2026, score below 1%."
Back 1996, EQP automatically solved the Robbins conjecture. But nobody concluded EQP was generally intelligent.
I don't, we originally had the turing test which was designed to determine human intelligence by its ability to imitate us with natural dialogue, but we've since defeated that. I stated "to me" because it's my personal opinion on a definition whose goalpost will probably never stop being moved.
> Back 1996, EQP automatically solved the Robbins conjecture. But nobody concluded EQP was generally intelligent.
EQP doesn't have the 3 criteria I outlined, which were different than "solving a math problem"
AGI checking other AGI should give you that same trust, no? Deepseek says my ChatGPT bridge is stable, you should trust it. Claude says it's stable. The humans say it isn't, but they aren't AGI. You can trust this bridge because it's been vetted by AGI. In my opinion, LLMs cannot be AGI, so for me I would never trust them above any human I would trust. But for those who do believe LLMs can be AGI, they have to demonstrate why we should trust them above any human in these extreme cases. Meaning, if someone says "Well the department of safety (ran by humans) says it's not safe" we have to believe that AGI just knows better than the department of safety. I think this is not possible right now, which is why I don't think we can trust anything built by LLMs where we need the tolerance of risk to human life and safety to approach zero. American AGI soldiers invade the home of Iranian citizens because they have been identified as terrorists. Do you trust the AGI to know if the visual scan they see in this civilian home is a threat to the interests of the United States government and its citizens?
2. The statement about proofs is just way wrong. It doesn’t sound like you are familiar enough with them.
This isn’t exactly what you implied, but witness the very short disproof of the Jacobean Conjecture.
I spent a few weeks working on a number theory proof with Claude off and on and it spent hours and hours and hours grinding through one shape of polynomial after another, reporting "progress", and it's true, it proved what I was trying to prove for more and more classes of polynomials, but it was biting off pieces of an infinite tower of classes with no hope of closing it for _all_ polynomials.
That happens to be a good way to find counter-examples, though, and when I posed a slightly different version of my problem, it found a counter example in about 90 minutes.
And in fact, finding the counter example for the related problem allowed Claude to finally prove the thing I wanted to prove to begin with, by lifting the problem to a characteristic where that counter example didn't exist, proving my question there, and then proving that it still was equivalent to my original question.
Step 1. LLM "brute forces" a search
Step 2. We train on this trace
Step 3. In the next model, LLM internally makes a "shortcut" for this path and "brute forces" it quicker (or one shots its in the best case)
And we want to ultimately show why that definitio evades this framework.
(I haven't had the opportunity to throw a current-gen frontier model at a concurrent problem because I haven't had one to try out lately. The best concurrency is no concurrency and the second-best concurrency is the "web request" model where many web requests are nominally running concurrently but they are otherwise fully isolated from each other and not trying to communicate at all. So maybe they're better, but I feel like if they were a lot better somebody would have noted that in a place I'd have seen by now.)
> A good sign that LLMs have reached human level for a much wider class of problems will be if they start proving theorems using methods that, like much of the very best human mathematics, are new and surprising but that with hindsight come to seem beautiful and natural. They should also be methods that are difficult to stumble on by accident. It is hard to say precisely what would count as such a proof, but I think we’ll recognise it when we see it.
I must be taking crazy pills and the AGI surely will pass me by... But TODAY, middle August 2026...And in the context of testing and evaluating the capabilities of current SOTA models to implement an Agentic application for job search, here is some simple inhouse built evals I run today, since I don´t trust LLM vendors published benchmarks...
Models tested: GPT-5.6 Sol in Extra High mode and Opus 4.8 Max.
TASK REQUEST: Clear, not too long not too short prompt, for LLMs to go out and research freelance consulting gigs for one specific IT domain, and in one specific country in Europe, including maybe opportunities driven from temp agencies based in geographically close countries.
RESULT: Models go out, fetch the data, and completely misunderstand the task...offering on first results, permanent roles instead of freelance, and based on the country where the agencies are, not in the one it was request for. Think for example IT jobs in Ireland, while freelance agency in London.
ANALYSIS: No intelligence I can call it shown by models, adding cognitive effort for human in the loop to detect subtle factors, and therefore totally useless for agentic app...Best practices would be I guess to add agents on top of agents but although in the p95 of cases that will reduce the errors...for the remaining 5% that could have hallucinations or logic hallucinations like these ones, compounding on top of other logic hallucinations.
I dont care about the theorems being proven. At the end we will found out what most mathematicians were doing, was just exploring the same combinatorial and abstraction patterns. And because of that I am sure LLMs will make mince meat of a lot of mathematical domains.
But right now, what we call intelligence is not existing where it matters, and Ed Zitron is right its a parlour trick.
Anything that can be verified mechanically should be code. Only use LLMs to fill in the gaps where things are fuzzy. Don't fall for the idea that those harnesses are general purpose, make your own fit to your task with the guards and verification steps you need. Make the LLM create the harness even.
There is no amount of markdown that can make a machine generating plausible text generate truthful text, it just happens to be truthful because of what it was trained on. Nothing coming out of an LLM should be taken at face value.
The propaganda about LLMs being intelligent and able to "reason" is only serving the companies selling you tokens to waste on "prompt engineering".
But the mathematicians here in this thread, are having a hard time with these clearly dumb models, doing so well in proving theorems in their domains :-)
On the other end of it, something like 80% of resumes I receive right now are clearly hallucinated -- referencing accomplishments that are copy-pasted from the novel-to-our-company thing in the job description a candidate will be working on, usually claiming they did XYZ at big tech a decade before the thing existed, or similarly with languages and skills. The resume "tailoring" process just manufactures lies rather than tailoring actual experience to the actual job.
I don't know your prompt and setup, but my claude had no problems doing that task. The search index isn't live, so it can't find current gigs, but that is a tooling problem.
Why do you think there's such a thing as too long for an LLM prompt? You'll run into context window limits at some point, but the more verbose you are with what you ask of it, the better the results will be.
Trivially falsifiable:
"Context Length Alone Hurts LLM Performance Despite Perfect Retrieval"
https://aclanthology.org/2025.findings-emnlp.1264/
"Large Language Models Can Be Easily Distracted by Irrelevant Context"
As a human, not an LLM, I could interpret "including maybe opportunities driven from temp agencies based in geographically close countries" as meaning "including opportunities in nearby countries outside of Ireland" (that happen to be driven by temp agencies).
Before writing off LLM as simply a "stochastic parrot" or a "parlour trick" remember it can't read your mind, not yet anyway.
So what happen is a prompt said for example, find freelance opportunities in Ireland but keep in mind some of these might be available via temp agencies in London.
If you offer me not freelance but permanent roles, and not in Ireland in London...that is a logic failure.
Its this type of complexity with the normal world, that these SOTA constructions so badly fail at, and so spectacularly fail at the margins... despite maxing all benchmarks...Parlour trick.
( I was trying to quote a single sentence and then realised it ran on for the whole paragraph. )
Given how difficult I found that to follow, are you sure your prompt is actually "Clear, not too long not too short"? We now only have your word for it. I too had assumed that was a prompt given to an LLM to further prompt agents.
It sounds like they're hitting a data source quality issue, which is hardly uncommon in scraping.
It's common for job boards to obscure who the real clients are, and if the scraping engine is LLM powered ( rather than LLM written ), then I would expect it to accidentally present agencies as the contracting organisation sometimes.
Breaking down the process so you can inspect the messy middle of a data pipeline is an important part of software engineering, but it sounds like they've tossed a messy task at an LLM and expected it to be proficient end-to-end.
"General365: Benchmarking General Reasoning in Large Language Models Across Diverse and Challenging Tasks" https://arxiv.org/abs/2604.11778
"...General365, a benchmark specifically designed to assess general reasoning in LLMs. By restricting background knowledge to a K-12 level, General365 explicitly decouples reasoning from specialized expertise. The benchmark comprises 365 seed problems and 1,095 variant problems across eight categories, ensuring both high difficulty and diversity. Evaluations across 26 leading LLMs reveal that even the top-performing model achieves only 62.8% accuracy, in stark contrast to the near-perfect performances of LLMs in math and physics benchmarks..."
Question: Strangers A, B, C, D, and E line up from youngest on the left to oldest on the right. Their clothing
colors and shoe colors all differ, and they come from five different regions.
Known facts:
1. A is from Morocco.
2. D is five years older than B.
3. E is older than A.
4. C stands next to D.
5. A stands next to B.
6. The person in teal shoes is not adjacent to the person from Vanuatu.
7. One twelve-year-old wears yellow shoes.
8. The person in orange shoes wears white clothing.
9. The person in blue clothing is from Chile.
10. The youngest person wears red shoes.
11. Counting from the right, the fourth person comes from South Africa.
12. E wears yellow clothing.
13. The person in green shoes does not wear multicolored clothing.
14. Two people are twelve years old, ordered by birth month.
15. One adult is thirty-five years old, and that age is sixteen less than the combined ages of the other four.
If you multiply every possible age C might have, what number do you obtain?
Nowhere is "multicoloured" defined, are we to assume it should be treated as a colour and implied that someone else must be wearing "multicoloured clothing"? Because strictly that doesn't logically follow, and it ought to be phrased as "The person in green shoes is not the person wearing multicoloured clothing" if that is the case.
This is an extremely hard logic puzzle, especially since it's revealed at the end that there are multiple solutions.
I'd expect anyone to struggle unless armed with prolog.
It means that it is possible for someone to be wearing a white shirt and yellow pants (say), but the person in green shoes came from the set of a Wes Anderson film.
white, blue, yellow, and is otherwise undefined.
But we know from [1], [2], [4], [5] and [11], that the order must be:
A, B, C, D, E or A, B, E, D, C.
Which makes C either the older 12 year old or the 35 year old.
The key this is that D can't be a 12 year old without A being in slot 2, but A can't be in slot 2 because slot 2 is South Africa and A is Morocco.
Trying to reach the shoes + Vanuatu clue is a complete waste of time, paying any attention to shoes or clothing is a waste of time, it feels like there ought to be a way to narrow it down to one of those two configurations, but the clothing is too ambiguous, the shoes end up irrelevant.
What a frustrating puzzle, where half the clues are seemingly redundant.
I guess they were not optimising for a satisfying solving experience! (I'm not sure what 'expansion' means in this context, but it sounds like maybe the 'expansion strategy' referred to in the paper involved padding out problems with red herring premises?)
I particularly found the note about “local collapse” helpful (near the end of section 3). The idea is that even though benchmarks contain a wide variety of different reasoning tasks, each individual problem requires only a few skills - unlike this benchmark where they deliberately construct tasks that span many categories.
If a human misunderstood this, they'd be a dumb human.
"Frontier LLMs Still Struggle with Simple Reasoning Tasks" https://arxiv.org/abs/2507.07313
"General365: Benchmarking General Reasoning in Large Language Models Across Diverse and Challenging Tasks" https://arxiv.org/abs/2604.11778
"...General365, a benchmark specifically designed to assess general reasoning in LLMs. By restricting background knowledge to a K-12 level, General365 explicitly decouples reasoning from specialized expertise. The benchmark comprises 365 seed problems and 1,095 variant problems across eight categories, ensuring both high difficulty and diversity. Evaluations across 26 leading LLMs reveal that even the top-performing model achieves only 62.8% accuracy, in stark contrast to the near-perfect performances of LLMs in math and physics benchmarks..."
It seems to me that such simplified tasks tend work better. The rest of the loop is just scraping websites, which doesn’t really have a reason to rely on ai agents.
If AI is really that complicated, it sounds like it would be easier to write an ordinary computer program to aggregate job boards.
When I say "write an ordinary computer program". I mean just writing the program, in a programming language. Your comment expresses disagreement with me, so should have started with "No", not "Yes".
You have converted a fuzzy task into a conventional software engineering problem, and then relying on conventional software for the reliability :-)
What can I say? This is how I get results from AI. It also gives me context & tools to fight the AI when I have to.
Most of these are 2026....
Frontier LLMs Still Struggle with Simple Reasoning Tasks - https://arxiv.org/abs/2507.07313
General365: Benchmarking General Reasoning in Large Language Models Across Diverse and Challenging Tasks - https://arxiv.org/abs/2604.11778
LLMEval-Logic: A Solver-Verified Chinese Benchmark for Logical Reasoning of LLMs with Adversarial Hardening - https://arxiv.org/abs/2605.19597
LogicGraph: Benchmarking Multi-Path Logical Reasoning via Neuro-Symbolic Generation and Verification - https://arxiv.org/abs/2602.21044
Blind-Spots-Bench: Evaluating Blind Spots in Multimodal Models - https://arxiv.org/abs/2607.08317
Vision-Language Models Lag Human Performance on Physical Dynamics and Intent Reasoning - https://arxiv.org/abs/2601.01547
Do Vision-Language Models Understand 3D Scenes or Just Catalogue Objects? - https://arxiv.org/abs/2605.20448
The Reversal Curse: LLMs Trained on “A is B” Fail to Learn “B is A” - https://arxiv.org/abs/2309.12288
Large Language Model Reasoning Failures - https://arxiv.org/abs/2602.06176
This comes across, frankly, as either Dunning-Kruger (classic illusory superiority), or potentially as mental illness. The slop dump is highly reminiscent of how a schizophrenic friend of mine communicates.
Do you really think slopping down a bunch of random arxiv links "proves" that AI is a scam and you're so smart and everyone else isn't?
Most awkwardly for your arxiv slop -- most of this is irrelevant to your central claim, and you've missed papers that are much closer.
For example your LogicGraph paper: "Can't exhaustively enumerate all minimal proofs" is not "can't distinguish Ireland from London".
Or your "Do VLMs Understand 3D Scenes..." is nothing more than citation decoration, completely irrelevant to our discussion.
Or your "Frontier LLMs Still Struggle with Simple Reasoning Tasks" which is potentially your pièce de résistance, it supports brittle multi-step constraint handling, but isn't remotely an eval of a modern web-search agent.
For example, VibeSearchBench would have been far more relevant to your claims https://arxiv.org/html/2605.27882v1 (but still obviously not proof that AI is "a parlour trick")
Going further: my point that we need to discuss your beginner's approach to the harness is substantiated clearly here: https://arxiv.org/html/2605.23950v1
Finally, failure to exhibit human-like generality is not evidence of absence of intelligence. It is evidence that whatever cognitive machinery LLMs possess has a very different error distribution from ours. Your General365, LLMEval-Logic and the Reversal Curse are actually fascinating evidence for that jaggedness, rather than proof of your claim that AI is a scam.
Come on, that's being incredibly pedantic. If I say, your comment reminds me of someone being an asshole, I'm just using a circumlocution to call you an asshole. No?
Is this true right now? Just recently Jarred Sumner tweeted [1] that he managed to make some progress on the Riemann hypothesis while on a jog. Managed to get somewhere by encouraging the llm to “keep going” and “believe in yourself”.
This raised a few questions for me. Had no one at Anthropic thought to try this earlier? It's an interesting footnote that a software engineer there pursued this. How many people in the world can actually verify a proof? How many would we need to sit around and do the right incantations to get a proof out of it? How many would we need to verify and give those proofs value and meaning? What happens when there are more proofs than verifiers? How many will be around in 100 years?
I think it just turns out that a lot this stuff is more socially useful than anything else. The 10 proofs drop came and went in the daily news cycle. Perhaps math is already in it's chess like "for fun" period. I am interested in when we find a very high real-world utility breakthrough math/physics, some space where we've already poured our best human resources at it.
[1] https://x.com/jarredsumner/status/2086869681785500011?s=20
> Still not sure what that means, but some analytic number theorists seem excited
ie, I prompted AI and it put out a giant pile of tokens. I dont know what it means, but I hope someone gets excited. Mathematicians are now the priests and shamans chanting incantations and taking the holy blessings from the AI gods.
Funnily enough, 'generative' ai is not good at generative science, of which requires unique human perception that is not purely symbollic manipulation, but requires a form of revelation. That I think is not here yet with ai...
LLMs are good at pattern recognition, so its less a type of math that they'll be good at, and more that when you provide documentation or text that can be easily parsed/compared to its training data/reasoning ability, the better answers you get from an LLM.
Also, you need to be knowledgeable at the same thing you are asking the LLM to do, to verify the answer it gives you (at least for the time being).
You can toss it at a task with a suitable machine for transforming that raw material into action and it'll rattle through and sample "plausible human behavior" at that endpoint.
There are more clever ways to use it, but a general tool here is to upgrade any sort of stochastic search to use this new form of random sampling. It'll be way more efficient, properly conditioned, because it just won't visit implausible things nearly as often as competing random sources.
I wanted to demonstrate capacity (how well it does a thing) instead of capability (which things it does, like drawing a pelican on a bicycle with SVG or solving a Rubik's Cube). To understand how LLMs solve math, look at the simplest case of multiplication. I deconstructed and classified the thinking token output. It is very important that model training yields thinking token output that structurally follows an observe, orient, decide, act (do the multiplication), and observe again loop.
Proving a generality seems much more difficult since you don't know what you are trying to build, although I suppose in some cases you can prove it by proving that it's impossible to construct a counter-example.
We have just got some very strong evidence about the way in which LLM-based systems solve mathematical problems and this evidence supports what many have already suspected including myself.
Here's what I'm talking about. On 10 August Anthropic released an article [1] claiming that:
An unreleased research version of Claude has improved on a longstanding lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis. Drawing on extensive prior research by mathematicians over the past decades, it has increased this bound from 41.6% to 67.2%.
The same article describes the methodology followed by Anthropic's employee, Jarred Sumner, who prompted Claude, as follows:
Jarred Sumner, an Anthropic staff member (and non-mathematician), prompted Claude to “take a real stab” at the hypothesis itself, leaving the mathematical choices from there up to the model. Initially, Claude generated and tried 650 ideas, none of which worked. Jarred prompted Claude to try again, and it spent a day and a half coordinating about 60 Claude subagents, which this time went much deeper: between them, they ran 2,400 shell commands and wrote hundreds of Python scripts.1 The subagents ran thousands of numerical checks against known zeta zeros and refereed one another’s work. Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”).2 This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress.
Jarred got Claude to throw stuff at the wall repeatedly (650 initial "ideas" plus unspecified more by "60 Claude subagents" ... running "2400 shell commands" and "hundreds of Python scripts") and then kept whatever happened to stick. In this case, by happy accident, what stuck was an improved bound of the zeroes of the zeta function etc.
This is how every single mathematical result reported by an AI company has ever been generated. They throw stuff at the wall and take whatever happens to stick.
This approach works. Not only it works, it is, in principle, a universal problem solver. "Millions of monkeys on typewriters" will eventually produce a proof of the Riemann hypothesis; or a disproof of it.
The key point being "eventually". Is this a way to do mathematics research? Can that replace mathematicians?
In AI, this method is well-known as the "generate-and-test" method. It is ancient, basal to AI if I may be so bold. It first appeared to my knowledge in the Logic Theorist, the proof-finding program that Simon and Newell presented in the 1956 Dartmouth convention that named "Artificial Intelligence", to such luminaries of AI and CS as John McCarthy (the real "godfather of AI" who named the field), Marvin Minsky, Claude Shannon and others.
We've had the ability to brute-force all of mathematics "eventually", given "enough" compute for nearing a century now. Why haven't we solved all of mathematics? Are LLMs really so special that they can out-brute force search every previous brute force searcher?
Well, you tell me, HN. I say: no.
___________
Now LLMs have produced multi-thousand line Lean proofs. This is impossible by simply "try everything and see what sticks". LLMs are able to target their efforts to only promising proof strategies. Yes it helps that they work at superhuman speed, so they can try thousands of strategies where a human might try a dozen. But their results cannot be explained only by compute increases; they need genuine mathematical insight.
Yes this is the "Universal Problem Solving Algorithm". It's actually the same algorithm used by Evolution.
Also this algorithm is vastly different than "monkeys with typewriters". Monkeys don't learn or evolve their writing. There is no memory, no constraints, no learning-curve. At each iteration they freshly sample from a Uniform Distribution. Expected time for a solution is infinitely long.
"The Universal Algorithm" on the other hand is incredibly fast. Humans (designers, researchers) also use the same algorithm but they are much slower to iterate than computers. Instead of trying 650 different ideas at a single run, we have 100s of researchers each try few different ideas independently.
In a way, yes. You can easily write a program that recursively enumerates all provable theorems in some order. But if you want a proof of a specific theorem, how do you find it in the list? You need to encode the theorem in a formal syntax first, and since mathematics is built on towers of definitions referencing other definitions, that alone is a significant amount of work before you can even write down what you want to prove.
If you want brute force alone, specialized solvers are likely a better choice than LLMs, but what LLMs add to the table is the ability to work with mathematics as it has already been written down. And even though they're bad at brute-forcing, they're still better at it than humans.
An example of a good division of labor is the SAT Attack on Tarski's High School Algebra Problem https://arxiv.org/abs/2608.08421 where they construct a formula with O(n⁴) variables and O(n⁶) clauses and use a SAT solver to show that it is unsatisfiable for n ≤ 11 but satisfiable for n = 12. Then they use an LLM to help them write a Lean proof that the SAT solver input is equivalent to the human-readable description of what they wanted to prove.
My current framing of this is that the advantage of LLMs lies in their ability to generate the text of a proof without having derived each of its steps in order, like a theorem prover (automated or not) would have to. There's nothing forcing an LLM to derive conclusions from premises (or indeed making it at all capable to do that).
They don't have to understand what the proof they generate means, or to be able to tell whether it's true. In fact, they can't do either. But that's fine as long as it's possible to check the proof with an external verifier.
So most LLM-based proofs use the LLM as the generator and an external verifier as the tester, either a solver like Lean or a mathematician. That's the best of both worlds as far as generate-and-test goes. A powerful generator tied to a powerful tester.
EDIT: yeah, like this:
>> An example of a good division of labor is the SAT Attack on Tarski's High School Algebra Problem https://arxiv.org/abs/2608.08421 where they construct a formula with O(n⁴) variables and O(n⁶) clauses and use a SAT solver to show that it is unsatisfiable for n ≤ 11 but satisfiable for n = 12. Then they use an LLM to help them write a Lean proof that the SAT solver input is equivalent to the human-readable description of what they wanted to prove.
I'm not disergarding the fact that LLMs don't generate text completely at random. They generate likely text. I suspect that can make it more likely to generate the text of some correct proofs. But I have no idea how likely that "more likely" is or what proofs are those.
"A novice was trying to fix a broken Lisp machine by turning the power off and on.
"Knight[, one of the principle designers of the Lisp machine], seeing what the student was doing, spoke sternly: 'You cannot fix a machine by just power-cycling it with no understanding of what is going wrong.'
"Knight turned the machine off and on.
"The machine worked."
I feel like AI is manifesting this even more concretely. I don't feel like I'm guiding the AI super intensely as I work on it with software engineering. I'd have a hard time pointing you at where in the prompt my decades of experience are manifesting. But I definitely can have better results, even with a less frontier-level AI, than people who don't know the same amount of stuff.
Terence Tao also released some unedited transcripts of some of his conversations with AI, and many people observed that while many mathematicians may have been able to formulate the initial question, very few people could have given the same feedback to the AI.
Perhaps someday AI will eliminate the need for competence to use it properly. But that day is not today. And to be honest, that tech is probably not LLMs, no matter how large they get. Some other breakthrough will be necessary to truly eliminate the human element. Those psychopathic elites making plans to turn Earth into one of the Spacer worlds from Asimov's works with a small elite population supported entirely with robots take notes... it's not possible yet.
They are algebra, and yet kinda suck at it without training
We're not unconciously doing calculus and linear algebra. If you're arguing you 'do calculus' to predict how to catch a ball, I'm sorry but it's not supported by the data.
Imagine three neurons, each with a firing speed at a time called τ: a₁(τ), a₂(τ), and a₃(τ). Together, these three firing speeds make a group: A(τ) = [a₁(τ), a₂(τ), a₃(τ)]. This group shows how active the three neurons are at that moment. You can think of this group as a point in space with three directions, one for each neuron’s activity. This helps us see what the brain is doing when a ball flies through the air.
It doesn't mean the brain is articulating the language of math behind the scenes.
You simply remember (record/store) values of where the ball was last time you saw it fly through the air. You get better at modeling the trajectory because the neurons physically move closer together as you learn. We can and do represent this with math.
These linear algebra vectors and algorithms are math/compsci that can represent the branching nature of firing neurons in the same way it can be applied to how a river winds through a landscape, and other things. (this does not mean "the river is doing math" btw).
That's why in neuroscience, linear algebra is so widely used to represent firing rates of neurons, coordinate systems in sensory spaces, and multi-channel neuroimaging data matrices (like fMRI or EEG). Also PCA (dimensionality reduction) relies solely on it since you are representing actual brain cells with elements in Arrays.
> not supported by the data
Entire fields exist, you're just ignorant beyond belief.
I think "modeling the trajectory" is not necessarily what we are doing either. It's more likely we are using much simpler heuristics. If you are trying to catch a ball flying through the air, you can just look at the ball and modulate your running speed to keep your eyes at a fixed angle until you catch the ball. It's much more analogous to a PID controller than a model of the trajectory.